It is well known that the Earth's subsurface can be imaged by a seismic survey. In a seismic survey, elastic waves generated at the surface of the Earth are injected into the Earth's subsurface by means of an explosive source such as a dynamite charge, in the case of a land-based survey, or an air gun, in the case of a marine survey. As the waves travel through the Earth, subsurface changes in the geological composition of the Earth cause the waves to reflect and propagate toward the surface. When the reflected waves reach the surface, they are recorded by a number of receivers. The receivers may be positioned at locations on the Earth's surface or towed behind seismic boats in the form of a very long cable made up of hydrophones. The data recorded by the receivers is processed to produce an image of the geological subsurface of the earth.
The data recorded by each receiver is a time series representing reflections from the subsurface caused by the explosive source at the surface of the Earth. This time series is known as a trace. The behavior of these traces is governed by the wave equation. It is known that by using the redundancy that is built into the method of data acquisition, the wave equation can be used to predict the speed at which the sound waves propagate through the Earth's surface. The wave equation predicts the dissipation of energy as a function of the known velocities of geological formations in the x, y, and z directions as the waves propagate through the Earth. After a large amount of data, in the form of seismic traces, is collected, it is processed. Following processing of the traces to eliminate noise, a stacking process may be performed in which traces are summed together into a three dimensional array of numbers comprising the amplitudes of reflected seismic waves recorded over a period of time. Following the noise elimination step, the stacked data can be migrated.
Data migration is the reconstruction of an image or map of the Earth's subsurface from the seismic data in the time domain as recorded by the seismic receivers at the Earth's surface. Data migration converts the data from the time domain to space or image-point domain. The data that exists in the time domain is mispositioned both laterally and vertically. Migration converts these mispositioned data to ones representing lateral and depth positions of geological structures. An example of seismic record migration is described in U.S. Pat. No. 6,021,094 to Ober et al, which is incorporated herein by reference. Although the step of data migration is often performed on post-stack data, migrating post-stack data can result in an imprecise or incorrect result. Simple stacking of traces can depict complex subsurface features in an incorrect location, or simple stacking of traces may have the effect of negating a subsurface feature in the stacked data. Because of the imprecision of migrated stacked data, pre-stack migration of the seismic records is preferred. The migration of pre-stack seismic records, however, involves a greater amount of time and computational resources, as compared to the migration of stacked seismic records.
The extra computation and time required for the pre-stack processing of data is well justified. The migration of pre-stack data results in the generation of migrated or image gathers. These image gathers are the fundamental diagnostics tools that are used to determine the correctness of the velocity model. The image gathers also provide the starting point for any velocity-updating scheme. The accuracy of a generated velocity model is dependent on the accuracy of the image gathers generated by the migration of pre-stack data. The difficulty of velocity-updating, whether accomplished by vertical or tomographic methods, resides in the fact that the ambiguity in the pre-stack data stems from the simultaneous arrivals of signals from a point in the Earth's subsurface to a given point on the Earth's surface. Wave equation migrations make this concern a moot point in that wave equation migrations are capable of collapsing the simultaneous arrival of signals into well-defined horizontal events in the migrated gathers, which in turn makes the velocity-updating procedure a more deterministic and robust one, capable of providing a more accurate and realistic answer. Moreover, amplitude versus offset (AVO) analysis also relies on migrated gathers, which depict geological events at a set of different offsets. The manner in which the amplitude of any particular subsurface events varies with offset depends on the property of the subsurface geology of the event. Such information aids geologists in better understanding the subsurface geologies in general and the location of hydrocarbon deposits in particular.
A number of algorithms have been used to perform the migration step. All are based to some extent on an approximation, simplification, or variation of the wave equation. Because of a lack of time and computational resource, the full wave equation, or the full solution to the depth imaging problem, has not been employed effectively to migrate pre-stack data. Employing the full wave equation would require both time and computational resources beyond what is now available or commercially feasible.